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It is known that when the $P_n(x)$ are the Hermite polynomials or standard Laguerre polynomials, the transformation $T$ has this property. It is also known that the transformation $T[x^n]=H_n^{\\alpha}(x)$, where $H_n^{\\alpha}(x)$ is the $n$th generalized Hermite Polynomial with real parameter $\\alpha$, has the differential operator representation $T[x^n]=e^{-\\frac{\\"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1707.05412","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2017-07-17T23:27:08Z","cross_cats_sorted":[],"title_canon_sha256":"08d58e1c07a796bafc6c37c8d02fc2b0f5746e741faaa100c48c4ad3e594f213","abstract_canon_sha256":"6824d1d3b8d329c7852af8ff71edec85d65d643c2ee0e819248d8d52108925e3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:02.418874Z","signature_b64":"kSYOApF/+BNtTzJ/OKn8cLXbvmm/O/0FnTeGH1JPXl8xYsMxg1l072wxGmFRYwzr8Y60zOw4JsGogH9hZ3g2CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2c5cfcc22d8420d91e22af024353cdc7926d6f1428fdcf53a2fbef6751d45e20","last_reissued_at":"2026-05-18T00:40:02.418406Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:02.418406Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Real-Root Preserving Differential Operator Representations of Orthogonal Polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"David A. 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